*V. Detailed Discussion of *

*Multiple-Pulse Sequences Intended for * *High-Resolution NMR in Solids *

With extremely few exceptions, researchers in high-resolution NMR in liquids at present buy their spectrometers from commercial manufacturers.

One of the questions a spectrometer salesman will invariably be asked by a potential customer is: What resolution can your instrument achieve? The answer the customer expects—and invariably gets without hesitation—is a number below 0.3 Hz.

Probably every experimenter who has made an effort in high-resolution NMR in solids by either multiple-pulse or magic-angle sample-spinning techniques has been confronted with the same question on repeated occasions.

Probably on no such occasion has he answered without hesitation. The reason is twofold (at least). First, he knows that his answer will be compared with something like 0.3 Hz, and that the number he will quote eventually will be much larger than that. Second, he feels compelled to explain that and why his answer cannot be a general one. It depends largely on the kind, size, shape, and orientation of the sample at hand. If he works with multiple-pulse sequences, he finally will give a number for a single crystal of CaF2 oriented with its 111 direction parallel to the applied field, and the number will be somewhere in the range of 15 to 100 Hz.

What we intend to illustrate with these remarks is

(1) Resolution is a problem in high-resolution NMR in solids and can be expected to remain one for the foreseeable future.

(2) In contrast to high-resolution NMR in liquids, where resolution is just

**91 **

**£ R P****7**** P P P P P****x**** % **

**A **

**A**

*■ cycle -*

FIG. 5-1. MREV eight-pulse cycle.

a matter of the homogeneity of the applied magnetic field,^{81} the problem is
very complex in high-resolution NMR in solids.

(3) The main efforts of advancing high-resolution NMR in solids are still done in research laboratories.

There is a large and still growing variety of schemes designed for achieving
high-resolution NMR in solids. However, only two seem to be currently in
use on a routine basis for actual data collection. These are the WAHUHA
four-pulse cycle, which has already served us for introducing the subject, and
an eight-pulse cycle first proposed^{82} by Mansfield^{83} and first applied success-
fully to CaF2 by Rhim et al.* ^{67}* This sequence is depicted in Fig. 5-1. We shall
discuss these two sequences in some detail. In doing so we shall encounter a
variety of aspects that bear on the practical usefulness of a line-narrowing
multiple-pulse sequence, e.g., resolution, scaling factor, signal to noise ratio,
sensitivity to misalignments of pulses, and observability of the NMR signal.

In Section F we shall briefly review further propositions for high-resolution NMR in solids.

For what follows it is useful to define ideal pulse sequences, which consist of pulses with width /w -> 0. Furthermore, the pulses are assumed to be perfect with respect to rf homogeneity, nutation angle, nutation axis, spacing, etc.

This enumeration may give already some impression of what kind of problems we have to deal with in practice.

**A. Properties of the Ideal WAHUHA Four- and MREV Eight-Pulse Cycles **
In Chapter IV, Section C,2, we discussed the lowest order or average
Hamiltonian approximation of the WAHUHA sequence. We stated that it
is adequate for t*c* ||^."^{ι}|| ^ 1· This condition, however, tells us nothing about

81 To be sure, we admire greatly the achievements of the NMR industry, which is now
able to offer magnetic fields homogeneous to within 1 part in 10^{9} over sample volumes as
large as about 1 cm^{3}!

82 Mansfield described this cycle in a shorthand but somewhat esoteric notation. This may be the reason why his fatherhood of the cycle remained unnoticed for some time. By the way, we shall not term pulse sequences with different preparation pulses as different.

83 P. Mansfield, /. Phys. C 4,1444 (1971).

**A. IDEAL WAHUHA AND MREV CYCLES 93 **

the line-narrowing efficiency of the cycle. Therefore, we must study the correction terms as they follow from the average Hamiltonian theory. In this section we shall evaluate the first- and second-order correction terms for the ideal WAHUHA and MREV sequences, but we shall also summarize those properties of these cycles that follow directly from the average Hamiltonian.

1. WAHUHA FOUR-PULSE CYCLE

a. Properties Related to the Detection of the NMR Signal

The initial amplitude of the oscillating part of the NMR signal is two- thirds the initial amplitude of a free-induction decay signal unless a special 45° preparation pulse is used. This means for typical applications of the WAHUHA sequence that the signal-to-noise ratio is substantially smaller than it could be.

There is a convenient 2τ window for sampling the NMR signal.

b. Properties of Zeroth-Order Average Hamiltonian

Scalar (zeroth-rank) /-spin-/-spin interactions remain unaffected. First- rank /-spin interactions are retained, but scaled down by a factor of 1/^/3.

The average of second-rank /-spin-/-spin interactions vanishes. The
smallest interval of time for which this average vanishes is 3τ = t*c**/2. *

c. First-Order Corrections

The cycle is symmetric (see Fig. 4-2); hence JF^{(n)}* = 0 for n odd, and in *
particular if^{(1)} = 0. This means the Magnus expansion

*F= f + f*^{(1)} + f^{( 2 )}+ ···

contains no purely dipolar terms quadratic in J^*O**, nor cross terms of the *
type Jif*O** 34?**cs* or Jf*D** Jif**of{*.

d. Second-Order Corrections

The evaluation of the second-order corrections Jf^{(2)} to the average Hamil-
tonian i f seems to be a formidable task in view of the threefold integrals
involved. In fact, it is not, at least not for ideal pulse cycles for which the
threefold integrals reduce to simple sums. For symmetric cycles such as the
WAHUHA cycle, further simplifications arise (see Fig. 5-2): From the sym-
metry of iF^{( 2 )} with respect to 3&(t*3**) and Jf (t^—they can be interchanged; *

see Eq. (4-41)—it follows that Ι*φ** = 7*φ and 7® = /φ, where I® is the contri-
bution of domain @ to Jf^{(2)}. This is indicated in Fig. 5-2. Hence it is sufficient
to restrict the integration over domains φ and (2). 7® (and 7φ) vanish, more-
over, when in addition to

*jfi(t) = 3&(t**c** -1) (condition a: symmetry of the cycle), *

**FIG. 5-2. Second-order average Hamiltonian for symmetric cycles. The symmetry of the **
**cycle and the invariance of **

i = .#<2> = zl [^{tc}*dh** [*^{t3}_{di2}* rdhtijrihU^hijrM]] + [jr(/i),[^(/*a),^(/3)]]}

**D l****c**** JO JO JO **

**against a permutation of ^(Ji) and 3^{t^) imply that the integrand of/is identical in volume **
**elements dV****Y**** and dV****2**** located at (J****u****t****2****,ts) = (a,b,c) and (t****c****-c, t****c**** — b****y**** t****c**** — a). If (0,6, c) is ****an arbitrary point in domain φ (domain @), (t****e****-c, t****c**** — b, t****c**** — a) is in domain ® (domain ****φ). It follows that 1****1**** = I**** 3**** and /**** 2**** = / 4 . **

we have

*jf = 0 (condition b: zero-average Hamiltonian). *

Proo/. For domain (2), i3 is not an integration limit of the integrals over
*t** _{x}* and t

*2*

*. Hence, the integration over t*

*3*can be carried out separately. Con- ditions (a) and (b) imply that J"£/2

*J&(t*

*3*

*) dt*

*3*

*= 0. Both conditions (a) and*(b) are met for the purely dipolar second-order correction term, given by

^^{2 > =}^{ 6 5 8 ^}^{2}^{[ ( ^ D}^{X}^{- ^ D}^{Z}^{) ,} W , ^ D^{y}] ] , (5-1)

where

*^D = y**n**2**hJ^U3cos*^{2}*!)**lk**-lHl/**r**l)(I*^{i}*.l*^{k}*-3I**z**%*^{k}*). *

**i<k **

*J^**D**X* and JF*D**y* are expressions analogous to J^*D**Z* with the indices z replaced
by x and y, respectively (see Table 4-2).

The damping constant associated with J ^^{2 )} is proportional to

*T**2,rigidC^.rigid/'c)*^{2}* where T2rigid is defined operationally, as before, by

<Δω^{2}>"^{1/2}. The coefficient of this lowest order nonvanishing correction
term,

**A. IDEAL WAHUHA AND MREV CYCLES ** **95 **
**tells us that the suppression of dipolar interactions starts to become effective **
**as soon as τ—not t****c****—becomes shorter than T****2rigid****. In view of the results of ****the early solid-echo experiments**^{84-87}** this is but natural—in retrospect. **

**An interesting point to note is that ^**^{2 )}** vanishes for so-called two-spin **
**systems.**^{88}** Otherwise stated, ^**^{2 )}** contains no terms proportional to (l/rf****k****)**^{3 }**but only terms proportional to (l/rf****fc****)**^{2}**(l/rf****i****), ΙΦ k, etc. In substances with **
**pairs of closely neighbored protons, e.g., methylene groups or water mole-**
**cules, the pair interatomic distance r****ik**** is considerably smaller than all other **
**interatomic distances. The absence of terms (\/rf****k****)**^{3}** in ^**^{2 )}** means that the **
**potentially largest terms, as far as geometry is concerned, actually do not **
**contribute to if^**^{2)}** and, hence, to the broadening of the spectral lines. **

**The offset second-order correction term is **

*** £ > = -Δω(Δα>ν/18){(/****χ****+/,+/****ζ****) + 3 ( /****ζ****- / , ) } . (5-2) **
**The rotation imposed on the spins by the (I****x**** + I****y**** + I****z****) part is along the same **

**axis as results from the zeroth-order term. It therefore modifies—slightly— **

**the scaling factor. Instead of 1/^/3 it now becomes (l/****x****/3)(l— Δω**^{2}**τ**^{2}**/6). **

**As the sampling rate in a WAHUHA experiment is \/t****c**** the maximum **
**frequency (Nyquist frequency v****N****) in the spectrum is v****N**** = \/2t****c**** = 1/12τ. **

**The frequency-dependent correction to the scaling factor therefore never **
**exceeds 4π**^{2}**·3/6·12**^{2}** = 0.138 = 13.8%. (The factor 3 enters because the **
**spectral frequencies are scaled frequencies, whereas Δω is the unsealed off-**
**resonance frequency.) Our experiments are usually arranged so that the **
**highest frequency of interest in the spectrum is below £v****N****, where the cor-**
**rection is about 3.5%. In our work on protons we even remain below £v****N****— **
**there the correction is negligible. **

**The rotation of the spins due to the (I****2**** — I****x****) part in Eq. (5-2) is perpendicular ****to the main rotation. It therefore affects the scaling factor only in second order. **

**However, it also tilts the rotation axis away from the rotating frame 111 axis. **

**The maximum tilt angle is 18.5°, which applies for the Nyquist frequency. **

**It is a matter of course that everything we have said about the offset second-**
**order correction term applies equally well to the chemical shift second-order **
**correction term. **

**Cross terms. It is clear that there is a multitude of cross terms involving **

**^cs> ^off> «*D>**^{ a n}**d «#j. The cross term quadratic in 3^****Ώ**** and linear in Jf****off****— **

**8 4****1. G. Powles and P. Mansfield, Phys. Lett. 2, 58 (1962). **

**8 5****1. G. Powles and I. H. Strange, Proc. Phys. Soc, London 82, 6 (1963). **

**86**** R. Hausser and G. Siegle, Phys. Lett, 19, 356 (1965). **

**87**** G. Siegle, Z. Naturforsch. A 21,1722 (1966). **

**88**** B. Bowman, M.S. Thesis, Massachusetts Institute of Technology, Cambridge, 1969 **
**(unpulished). **

presumedly the most important one—is given by

(τ^{2} Δω/18){2[JfD*, [JTD», / J ] + 2 |>rD«, [JTD», / J ]

- [jfD*. [JfD*,/J] + [/„[JTD*, JfD*]]}. (5-3) The damping constant associated with this term is proportional to

**T**** (T****2****, rigid _ J _ \ **

Y 2'^{r i g i d}^ τ ' τ Δ ω /

Note that the damping increases linearly with the offset! We avoid writing expressions for other cross terms because they can be expected not to limit the resolution in actual experiments.

2. MREV EIGHT-PULSE CYCLE

This cycle was first described by Mansfield^{83} as one of an entire family of
eight-pulse, 12τ complementary cycles that compensate for both finite pulse
width and rf inhomogeneity effects. Rhim et al.* ^{67}* first used it in 1973 in actual
experiments and got amazingly good line-narrowing results. Later it turned
out, however, that the improvement resulted to a small extent only from the
particular sequence and more so from an overall improvement of the apparatus.

We describe here the properties of the ideal MREV eight-pulse cycle insofar as they differ from those of the WAHUHA cycle.

a. Properties Related to the Detection of the NMR Signal

The initial amplitude of the oscillating part of the NMR signal can be made equal to the initial amplitude of a free-induction decay signal by applying a properly chosen 90° preparation pulse (see Fig. 5-1). The maximum potential signal amplitude can thus be exploited fully with ease. The oscillating signal does not ride on a dc pedestal as it does with the WAHUHA sequence. The absence of the pedestal is of particular importance when spin-lattice relaxa- tion and/or (weak) IS dipolar couplings cause in WAHUHA experiments a decay of the pedestal, which is considerably inconvenient for the subsequent processing of the data.

b. Properties of Zeroth-Order Average Hamiltonian

First-rank /-spin interactions are retained, but scaled down by a factor of V^/3, which is smaller than the corresponding WAHUHA scaling factor.

c. First-Order Corrections

The total MREV eight-pulse cycle is not symmetric; hence, Jf^{(1)} does not
vanish identically. With the aid of Fig. 5-3 it is easily verified that the offset
(Δω) and chemical shift (Δω,·) first-order correction terms are given by

■*&> + ^ = - * τ Σ (Δω + Δω,)^{2}*(Ij-lj). (5-4) *

A. IDEAL WAHUHA AND MREV CYCLES 97

**0 3T 6T 9T f****c **

FIG. 5-3. First-order correction term J^^{(1)} for MREV eight-pulse cycle.

J?cs(0 + - ^ f f ( 0 ° C /e.

Domains φ and ® encompass symmetric subcycles and do not contribute to Jf^{(1)}. Domains
*2χ - 2*4 are equivalent. <^02) and ^ ( / i ) commute in cross-hatched areas. Contributions
from areas marked by crosses cancel.

The first-order dipolar correction term vanishes: Domains φ and ® in Fig.

5-3 encompass symmetric subcycles that do not contribute to if^{(1)}, and
domain (2) involves the sum Jf*D**x* + J4?*D**y* + J^*O**Z** = 0. The same arguments *
imply that the first-order dipolar-offset and dipolar-chemical-shift cross
terms vanish.

d. Second-Order Corrections

The purely dipolar second-order correction term is identical with the
corresponding WAHUHA term if it is written in terms of τ instead of t*c*.
There are again cross terms between ^fD, J>f*cs**, ^f*off, and JfJ. The coefficients
are of the same order of magnitude as for the WAHUHA sequence.

From this enumeration the following conclusions may be drawn:

(i) The ideal (later) MREV cycle is hardly superior to the ideal (earlier) WAHUHA cycle. In Section D, where we consider pulse imperfections, it will become obvious that the MREV cycle is superior to the WAHUHA cycle.

(ii) The resolution "should" be best close to resonance since the second-
order cross terms between Jf*O* and J«foff vanish at, and are very small close to
resonance.

(iii) <#£^{2)} appears to be the resolution-limiting factor.

Experimentally it is found in agreement with (ii) that the resolution deteriorates
far off resonance^{89,90}; however, since our early multiple-pulse experiments

89 W. K. Rhim, D. D. Elleman, and R. W. Vaughan, /. Chem. Phys. 59, 3740 (1973).

90 A. N. Garroway, P. Mansfield, and D. C. Stalker, Phys. Rev. B 11, 121 (1975).

on CaF24 6 , 5 1 it became clear that the resolution tends consistently to be better somewhere off, rather than exactly on resonance. Later we realized that this is due to a further averaging process, which becomes operative off resonance and which we are going to describe now.

**B. Off-Resonance Averaging **

We start by recalling that the suppression of homonuclear dipolar couplings discussed so far is the combined result of two averaging processes:

(i) The application of a strong magnetic field Bst imposes a fast common
motion on the spins, which makes it possible—even mandatory—to truncate
the full dipolar Hamiltonian. We are left with the truncated or secular dipolar
Hamiltonian, which is just the time average of the time-dependent full dipolar
Hamiltonian in the rotating frame. There was and is no need at this stage to
consider corrections ÜF^{(1)}, «#^{(2)}, etc., to the average Hamiltonian because
the Larmor period—which plays the role of the cycle time—is typically
smaller by three orders of magnitude than T*2 rigid* in very low, in principle in
zero, applied field.

(ii) The application of any one of the line-narrowing multiple-pulse
sequences swirls all spins synchronously around and leads to a zero average
of the truncated dipolar couplings in the toggling frame. Because the cycle
time of the multiple-pulse sequence typically cannot be made much smaller
than T*2 rigid* in high fields we had to consider correction terms to the average
Hamiltonian. The lowest order nonvanishing purely dipolar correction term
is ^^{2}> for both the WAHUHA and MREV cycles.

The effective Hamiltonian in the toggling frame with the applied field set somewhat off resonance consists of

(i) a zeroth-order resonance-offset term, the detailed structure of which depends on the particular pulse sequence at hand,

(iii) a host of cross terms,

(iv) a host of pulse imperfection terms.

All these terms have equal rights in the effective Hamiltonian irrespective of whether they arise just as averages or as higher order correction terms.

In the toggling frame the resonance-offset term imposes a common uniform
motion on all spins of a given isotopic species. It is just a third repetition of
our by now familiar game to treat this common motion of the spins by a further
interaction representation, U*off*. The consequence is that the remaining parts

**B. OFF-RESONANCE AVERAGING ** 99
of the toggling-frame effective Hamiltonian—^^{2 )}, cross terms, pulse im-
perfection terms—acquire a time dependence, e.g.,

Over this time dependence we may average eventually—provided it is fast enough.

*U**of{* represents a rotation

(i) about the toggling-frame 111 direction for the WAHUHA four-pulse sequence,

(ii) about the toggling-frame 101 direction for the MREV eight-pulse sequence.

It is over these motions that we must average the time-dependent toggling- frame Hamiltonian. (Recall that the toggling and rotating frames can be considered to coincide if the spin system is observed stroboscopically.)

What is the result of this third averaging process? First of all, the time
average of UF^^{2)}(0 vanishes for the WAHUHA four-pulse experiment! It
is possible, but rather tedious, to demonstrate this for a continuous rotation
about the 111 toggling-frame axis (see Appendix C). It is much easier—and
still sufficiently instructive—to consider a stepwise rotation with three steps
for one full turn. This we shall do now.

We expressed iF£^{2)} in terms of ^Dx, jT*O**y**, and Jf**D**z**. The structure of 3tf**D**x *

etc., is evidently such that each 120° rotational step about the 111 axis carries successively

*x -> y, y -> z, z -> x. *

The averge of J ^^{2 )}( 0 becomes

+ [(XJ-X&IXJ,*^}. (5-5)
A consequence of 3V*a**x**+3V**O**y**+3tf**O**z* = 0 is

The inner commutators in Eq. (5-5) can therefore be bracketed out and the
terms of the remaining factor cancel. Hence we have the result that the average
of Jp£^{2)}(0 vanishes for the WAHUHA sequence.

Off-resonance averaging substantially reduces but does not eliminate
completely JF£^{2 )} for the MREV eight-pulse sequence. The direction of the
off-resonance rotation axis is not as favorable as for the four-pulse sequence.

**Let us ask now: How important is off-resonance averaging in practice? **

**This question is, in fact, somewhat too general. Therefore, let us ask in more **
**detail: **

**(i) What about experiments that clearly demonstrate the effectiveness of **
**off-resonance averaging? **

**(ii) What consequence has off-resonance averaging for the design of **
**highly efficient line-narrowing multiple-pulse sequences? **

**(i) Simple, rather than sophisticated, multiple-pulse sequences are best **
**suited for demonstrating the effectiveness of off-resonance averaging. An **
**example is the phase alternated tetrahedral angle (PAT) sequence**^{91}**'**^{92 }**sketched in Fig. 5-4. **

**On resonance, the PAT sequence does not average out the dipolar couplings **
**between / spins. In fact, no pulse sequence consisting of two rf pulses only **
**can do that.**^{51}** Indeed, in an experiment on resonance on CaF****2****, we could **
**lengthen the decay of the**^{ 19}**F NMR signal barely by a factor of two. **

**Off-resonance averaging theory predicts a complete suppression of the **
**/-spin dipolar couplings,**^{91}** and indeed a more than threefold stretching of **
**the decay—as compared with the on-resonance case—was observed on **
**going off resonance by 8 kHz.**^{91 }

**Pines and Waugh**^{93}** realized that the stretching of the decay in this particular **
**experiment was limited by a finite pulse-width effect, and that it could be **
**overcome by adapting the nutation angle to the duty factor 2/****w****/i****c**** of the **
**sequence. These authors report having attained with the PAT sequence a**^{ 19}**F **
**decay time in CaF****2**** in excess of 1 msec. They also give further examples of **
**simple pulse sequences with marked off-resonance averaging effects together **
**with a very intricate theory. **

**(ii) While there is no doubt that off-resonance averaging helps improve **
**resolution with WAHUHA four- and MREV eight-pulse sequences there is **

**p/09.5 pH)9.5 pKJ9.5 **

**time **

**T **

**- cycle -**

**zr **

**zr**

**FIG. 5-4. The phase alternated tetrahedral angle (PAT) sequence. P is a preparation **
**pulse and may be just another PI**^{0}*****^{5}** pulse. **

**9 1**** U. Haeberlen, J. D. Ellett, and J. S. Waugh, / . Chem. Phys. 55, 53 (1971). **

**9 2**** J. D. Ellett, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, 1970 **
**(unpublished). **

**9 3**** A. Pines and J. S. Waugh, / . Magn. Resonance 8, 354 (1972). **

**C. T|| AND T****±****; HETERONUCLEAR DIPOLAR COUPLINGS 101 **

doubt that for the four-pulse sequence the improvement results predominantly
from the suppression of «#^^{2)}. We are led to this suspicion by the fact that the
MREV eight-pulse sequence—where iF£^{2)} is not fully eliminated by off-
resonance averaging—seems to yield significantly better resolution than the
four-pulse sequence, where ^ ^{2 )} is fully eliminated.

A side conclusion from this observation is that there is no practical point
now to use sophisticated multiple-pulse sequences for which not only
ifD = jpv-) = 0, but also JF^* ^{2)}* = 0. The first such sequence has been proposed
by Evans,

^{94}and Mansfield

^{83}has devised a whole family of sequences that possess this property.

So, what gets eliminated by off-resonance averaging in WAHUHA and, above all, in MREV experiments? We get a hint by noting that the amount by which the field has to be set off resonance in order to attain the best resolution tends to become smaller and smaller as experimenters improve their equip- ment more and more. This indicates that off-resonance averaging is also effective in suppressing adverse pulse imperfection effects. With "poor"

equipment this is probably the most important off-resonance effect in MREV and WAHUHA experiments.

We shall study pulse imperfections rather carefully in Section D, but first we shall consider another interesting resonance offset property of multiple- pulse sequences, namely, a dramatic difference in relaxation rates parallel and perpendicular to the effective field in the toggling frame.

**C. T|| and Γ**±; Heteronuclear Dipolar Couplings

Consider Fig. 5-5, which is taken from Haeberlen et al.* ^{91}* It shows the

1 9F NMR signal of CaF2 in an on- and an off-resonance WAHUHA experiment.

On resonance, the magnetization <M> decays nearly to zero over the time
of the WAHUHA experiment, off resonance it does not. The oscillations
almost do, but not the pedestals. As we have seen in Chapter IV, the pedestals
arise from the component of <M> that is initially parallel to Beff in the toggling
frame, whereas the oscillations arise from the component of <M> that is
perpendicular to Beff. For convenience we shall characterize by time constants
T± and T|| the decays of <M>, respectively, perpendicular and parallel to Beff,
even though these decays need not be simple exponential. T*L* corresponds to
the usual WAHUHA decay time. That two relaxation rates are involved in
NMR line-narrowing experiments had already become evident from the work

**W. A. B. Evans, private communication (see Haeberlen and Waugh**^{51}**)· **

FIG. 5-5. WAHUHA experiment performed on a single crystal of CaF2. Horizontal
scales = 1.0 msec/division. Upper trace, on resonance; lower trace, 2.0 kHz off resonance
(from Haeberlen et al.^{91}*). *

of Lee and Goldburg^{61} and Tse and Hartmann.^{95} In both of their experiments
solid samples were irradiated with a strong rf field whose intensity and
frequency offset from resonance were chosen to produce an effective field in
the rotating coordinate frame, which made the magic angle with the z axis.

Lee and Goldburg measured the rate of decay (Τ±_1) of the component of

<M> perpendicular to the effective field, while Tse and Hartmann studied the
relaxation (T^) of the magnetization along Beff and they found Γ|( to be orders
of magnitude longer than T*L* as found by Lee and Goldburg.

95 D. Tse and S. R. Hartmann, Phys. Rev. Lett. 21, 511 (1968).

**C. Γ|| AND Τ****λ****; HETERONUCLEAR DIPOLAR COUPLINGS 103 **

In one experiment we measured T*{1* in an off-resonance WAHUHA experi-
ment on a single crystal of CaF2 doped with paramagnetic U* ^{4+}* ions and found
it to exceed 0.6 sec, which is two orders of magnitude greater than its on-
resonance value. (A Ty of 0.6 sec exceeds the spin-lattice relaxation time 7i
of that crystal. This is possible because in the WAHUHA experiment, as in
the experiment of Tse and Hartmann, the spin diffusion relaxation mechanism
is suppressed.) T

*±*only increased by a factor of roughly three on going the same amount off resonance.

We can use the resonance offset-averaging theory to understand in a
qualitative way the difference between the resonance offset dependences of
T|| and T*±**. The on-resonance decay of the magnetization is determined by *
iP^^{2)} and pulse imperfections. We have seen above that off resonance we must
average these correction terms over the motion caused by Beff in the toggling
frame. As in Chapter IV, Section C,4,d, it is again helpful to introduce a new
set of axes in the toggling frame, the Z axis of which points along the (old)
111 direction. The average of a typical correction term if|^{n)} over the motion
generated by Beff has the form

<^{iP(}"^{>}>- = γ*π** j^expWV&ptxpi-MIz)**. *

This expression commutes with 7Z, which we can easily see if we express JFj^* ^{n) }*
in terms of spherical tensor operators T

*lm*to take advantage of their simple transformation and commutation properties.

<c#A(n)>av can be written as a sum of terms of the form

*Γ\χρ(ίΦΙ**ζ**)Τ**1ηί* exp(-/0/z) άΦ = — j \xp(imQ>)T*lm** άΦ = S**m0**T**l0**. *
But[/z,Tz o] = 0.

Therefore, back in the old toggling frame (or rotating frame if observation of the spin system is restricted to the proper windows) we have

**[ < ^ i**^{M )}**> a v , ( / x + / , + / z ) ] = 0 . **

Thus, for a sufficiently large resonance offset the correction terms to the
on-resonance average Hamiltonian are replaced by terms that commute with
*(I**x** + I**y** + I**z**). It follows immediately from this and the equation of motion *
(Heisenberg equation) for the magnetization operator M ( 0 ,

**tö(0 = '{(lI<^i'**

^{,>}**>av+-),M(/)l, **

that the time derivative of the component of the magnetization parallel to the 111 direction in the toggling frame is zero. [The dots indicate higher order

1

*2K *

corrections to the present averages.] This accounts for the fact that in multiple- pulse line-narrowing experiments performed off resonance the components of

<M> parallel to the 111 direction decay much more slowly than the com- ponents of <M> perpendicular to that direction.

On resonance, T± is determined by a number of correction terms, some of
which commute with (I*x**+I**y**+I**z**) and some of which do not. Off resonance, *
only terms that commute with (I*x** + I*_{z}* + I*_{y}*) remain to first order, so that T*±

increases somewhat off resonance, but not so much as Ty, which is not affected
at all by these terms. Since the average chemical-shift Hamiltonian jf*cs *

commutes with (I*x** + I**y** + I**z**), the Heisenberg equation of motion above implies *
that the component of <M> along the 111 direction will not carry information
about the chemical shifts of the sample. Thus we see that T*l9* rather than Τ|(,
gives a measure of the chemical shift resolution of a multiple-pulse line-
narrowing experiment.

In crystals containing more than one species of nuclear spins the WAHUHA
T|| is much greater than T*L* both on and off resonance. Figure 5-6 (again from
Haeberlen et al.^{9i}*) shows the WAHUHA*^{ 19}F decay of a single crystal of
LaF2. Notice that there is a rapid initial decay of the signal train to « ^ of
its initial value, followed by a much slower decay. The initial decay time is
48 //sec, which is longer than the 20.2 /zsec^{ 19}F free-induction decay time of
the crystal at this orientation. The slowly decaying signal never develops a
beat structure as the spectrometer frequency is shifted away from resonance,
although the time constant of the decay decreases as the resonance offset is
increased beyond several kilohertz. The difference between T^ and T*±* in this
experiment follows from the nature of the truncated heteronuclear dipolar
interaction Hamiltonian, ^s^ifar, which in this experiment dominated the
decay.^{96} It behaves in the /-spin space as a first-rank tensor so its average
during the pulse sequence, both on and off resonance is

**« f , r = iy**

**a**

**I****y**

**y**

**n**

**s****l>T**

**l>T**

**k**

**3****V-3cos**

**V-3cos**

^{2}**!)**

**!)**

**ik****)s**

**)s**

**z**

**i****(i**

**(i**

**x**

**k****+i**

**+i**

**y**

**k****+i**

**+i**

**z**

**k****). **

**).**

**i,k **

As usual, the /-spin species (^{19}F) is the one we observe and hit by the multiple-
pulse sequence. This average Hamiltonian commutes with (I*x** + I**y** + I**z**) and *
as noted earlier, this implies that the decay of the component of <M> along
the 111 direction will be slower than the decay of the component of <M>

perpendicular to that direction. Thus T^ should exceed T*l9* as indeed is the
case in Fig. 5-6. The resonance offset average Hamiltonian commutes with
*(I**x** + I**y** + I**z**), which accounts for the fact that no beat structure was observed *
on the slowly decaying signal.

**96**** Suppression of ^RciL· by either heteronuclear or self-decoupling has been discussed in **
**Chapter IV, Section F,5. **

**D. PULSE IMPERFECTIONS ** **105 **

1.00

0.80

§ 0.60

**< **

t 0.40

**CD ****< **

0.20

"'O.O 1.0 2.0 3.0 4.0 ms

**FIG. 5-6. WAHUHA experiment performed on a single crystal of LaF****2**** (from Haeberlen **
**etal**^{91}**). **

**D. Pulse Imperfections **

Naturally we can never excite our spin systems with ideal pulses in real experiments. Since the averaging achievable with ideal pulse sequences—

which we have considered thus far—appears to be fantastically good, and since the results of the pioneering experiments in all multiple-pulse laboratories were or are not that good, suspicion has arisen that the discrepancy is due to pulse imperfections. We are aware of a host of pulse imperfections and there may be more we are not aware of. For most of the pulse imperfections that we understand, compensation schemes have been worked out. We shall discuss them in this section.

The following pulse imperfections have attracted interest thus far:

finite pulse widths,

flip angle errors common to all pulses, rf inhomogeneity,

power droop,

flip angle errors different for the different sets of pulses, phase errors,

phase transients.

We shall consider here another conceivable imperfection, namely, timing errors.

1. FINITE PULSE WIDTHS; COMMON FLIP ANGLE ERRORS;

rf INHOMOGENEITY; POWER DROOP

These pulse imperfections form a class to which we shall turn first. Their important common feature is that they do not destroy the cyclic and the

symmetry properties of multiple-pulse sequences (see below). For a power
droop this is true only in so far as the power droop over individual cycles can
be neglected. Our approach will be completely straightforward: We consider
the average Hamiltonian of spin systems subject to multiple-pulse sequences
composed of pulses of finite width that flip the nuclear magnetization through
variable angles ß. We shall learn how the zeroth-order effects of finite pulse
widths can be overcome.^{97} We shall further learn to understand the sensitivity
of various pulse sequences to flip angle errors common to all pulses and thus
also to a droop of the rf power in the course of a long pulse train. Our approach
also provides an understanding for the effects of rf inhomogeneity since rf
inhomogeneity means nothing but a distribution over the sample volume of
common flip angle errors. Again we shall start by considering the WAHUHA
cycle. The (superior) properties of the MREV eight-pulse cycle can easily be
inferred from those of the WAHUHA four-pulse cycle.

As this class of pulse errors does not destroy the symmetry properties of multiple-pulse sequences, the WAHUHA sequence remains symmetric even when the pulses have a finite width. As a result all corrections of odd order to the average Hamiltonian still vanish automatically (see Chapter IV, Section D,4). We think it is reasonable to suspect that second-order corrections, and higher order corrections of even order in general, are modified only slightly by pulse imperfections—which we are trying to keep down anyhow—so it is not worthwhile going through the trouble of studying them for imperfect pulse cycles.

a. WAHUHA Four-Pulse Cycle with Pulses of Finite Width;

*Condition for Jf*D = 0; Scaling Factor

Figure 4-2 shows the timing of a WAHUHA sequence with pulses of finite
width. Also shown is the evolution of the propagation operator U*Tf**(t), which *
is evidently symmetric about the midpoint of the second large window. The
last two columns of Fig. 4-2 are intended to provide a "look and see" proof
of that fact. Since U*rf**(t) is symmetric, J4?j*^{nt}*(t)= U~**{** 1** (t)^**s**l**^**ulaT** U**T{**(t) is *
also symmetric. The consequence of this property of ^^{nt}*(t) with regard to *
*JF** ^{(1)}* has been mentioned above. Another consequence is that for evaluating
i f it is sufficient to consider only one half of the cycle. The average of
-ocular (0>

^{ m}particular, is proportional to

1 ^

**r. i* - 3</**

**z**

**f**

**(o/**

**z**

**k**

**(/)>**

**av**

** = r. i* - - 2 3</;(0//(0V**

**P**

**p=l **

where the summation is over the intervals p = 1,..., 5 of the first half of the
WAHUHA cycle; <···>ρ means average over interval p\ t*p**is the duration of *
interval p.

**M. Mehring, Z. Naturforsch. A 27, 1634 (1972). **

**TABL****E**** 5-****1 ** **AVERAGE****S O****F**** Ij****(/)//(/)**** OVE****R**** INTERVAL****S /?****=**** 1,...,****5**** O****F**** WAHUH****A**** SEQUENC****E**** (SE****E**** FIG****. 4-2)****a ** **Coefficien****t o****f ** **5 τ **

**/,'/** W**** h%****k**** i(h%****k**** + VA*)**** i(h%****k**** + W) «/,'/,* + /,'/,*) **

**~ :** ** ^** ** :** ** :** ** r** **-**

**/w**

**/ sin2jff**

**\ /**

**w**

**/ sin2^**

**\ sin**

**2**

**jff**

**—**

**τ(ΐ--)**

**δ**

**ΐη**

**2**

**£**

**T(I--)COS**

**2**

**A**

**-**

**-**

**τ(ΐ--)**

**δ**

**ΐη2**

**£**

**/w/**

**, sin2jff**

**\ /**

**w**

**/**

**ύη2β\****.**

**.**

**/**

**w**

**/ si**

**n**

**2β\****,**

**Λ**

**sin**

**3**

**A**

**sin**

**2**

**jft**

**Λ**

**/**

**w**

**/,**

**si**

**n 2jff**

**\ .**

**Λ**

**Λ**

**(l-^jsin**

**2**

**^**

**dl-Mcos**

**2**

**£sin**

**2**

**jf**

**f τΠ-^Jcos**

**4**

**^**

**τί****1**

**-**

**^jsin2Asin**

**£**

**T**

**{ 1**

**-**

**^\sm2ßcosß****τί**

**ΐ -**

**^)sin2£cos**

**2**

**£**

**fl**

**Th**

**e coefficient**

**s o**

**f th**

**e variou**

**s operato**

**r form**

**s o**

**f**

**Ι**

**ζ**

**ι**

**(ί)Ι**

**ζ**

**κ**

**{ί)****multiplie**

**d b**

**y**

**t**

**p****ar**

**e shown**

**.**

**t**

**p****is**

**th**

**e lengt**

**h o**

**f interva**

**l**

**p.**It is straightforward to evaluate the averages (Ι*ζ**(**(ί)Ι**ζ**Η**(ί)}**ρ**. The heading *
in Table 5-1 shows which operator forms are generated by U*Tf** (t) from I**Z**I**Z**. *
The further entries of the table show the coefficients, multiplied by t*p**, of these *
operators in the averages (I*z**(**(t)I**z**k**(t)y**p**. *

The average of ^s«uiar(0 contains, of course, the same spin operator combinations as does the heading in Table 5-1. The coefficients of these operators in the average of «^c uia r(0 are given in Table 5-2. Starting from Table 5-1, only simple juggling with trigonometric functions is required to find the expressions of Table 5-2.

The outstanding feature of Table 5-2 is that all the coefficients have one factor in common. Suppression of dipolar interactions is evidently obtained when this factor becomes zero, which means when

In Eq. (5-6) let us consider ί*ψ**/2τ as a parameter and the flip angle β = ω**ί**ί**γ/ *

as a variable. Note that this assignment of roles follows closely at least a
possible procedure of alignment of actual multiple-pulse sequences. The
range of ί*ψ**/τ is clearly 0 ^ r*w/ r ^ l . The equality signs apply only for
idealized cases. By a different approach, which did not state the problem as
clearly as do Table 5-2 and Eq. (5-6), we recognized very early^{51} that ^s?c u l a r

can be suppressed not only for iw = 0, but also for small but finite values of
the parameter fw/r. However, it was Mehring^{97} who realized—still using a
somewhat different approach—that there is a choice of β for the entire range of

TABLE 5-2

AVERAGE OF Γ · Ι * - 3 /Ζ7Ζ* OVER W A H U H A FOUR-PULSE CYCLE WITH PULSES OF FINITE WIDTH /wa

Operator Coefficient

*IJU *

**Ι,Ί," [ - ] COS**^{3}**£ **

/Z7Z* - [ . . ] ( l + cos^{2})9)cos^

*Wx%*^{k}* + h%*^{k}*) - [ - ] 2sin*^{2}£
*HW+UIS) -[···] 2 sin 2ß *
i(/„7x*+/,7/) -[...](l + cos^{2}£)2sin)ff

*a* The coefficients of the relevant spin operator combina-
tions in <I'-I*—3/z7zk>av are shown. The expression in the
square brackets, [···], is the same in all coefficients.

**D. PULSE IMPERFECTIONS ** 109
*tyf/τ that satisfies Eq. (5-6). We shall denote the particular value of the flip *
angle ß that satisfies Eq. (5-6) by ß*0**, which is a function of **W/T.

For tyf/τ = 0, ß*0* is, of course, equal to \π. For ί*ψ**/τ = 1, Eq. (5-6) leads to *
the transcendental equation tan/?0 = — β*0**, which has β**0** — 116.24° as a *
solution. For intermediate values of JW/T, β*0* varies monotonically between
these two limiting values. Mehring^{97} gives a plot of β*0* versus f/w/r (which is
the duty factor of the pulse train).

In our experiments we usually select a certain value for *W/T, typically 0.2.

Then we adjust β for optimum resolution. If everything is aligned properly we indeed obtain the optimum resolution for β slightly larger than \π.

In summary, homonuclear dipolar interactions can be suppressed up to and including first-order corrections in the Magnus expansion by WAHUHA four-pulse sequences with pulses of finite width. The condition is that the flip angle β is chosen such that it satisfies Eq. (5-6) for the selected ratio of fw/τ.

Finite pulse widths affect not only the suppression of dipolar interactions,
but also the scaling factor and, weakly, the off-resonance averaging mech-
anism. To study their influence on the scaling factor we must consider the
average of I*z**(t). This is easily obtained with the aid of Fig. 4-2. We give *
only the result:

</,(0>.v

= <^I7^{1}(0/,t/r f(0>.v

1 Ι * Γ · R 'w/sinj? 1-cosjgyi

+ /, (l+cosj3+cos^{2})3)

*tv/f * _ l + c o s^{2}ß* /t 0**^*^{sin}*ß *

cosj5 + ^—£ - (1 +cosj?) —f (5-7)

Denoting the coefficients of I*x**, I**y**, and I**z* in Eq. (5-7) by C*x**, C**y9* and Cz,
respectively, the scaling factor S may be expressed by

*S = KC**X**2** + C**y**2** + C**z**2**)*^{112}*. (5-8) *

With the aid of any decent pocket-sized calculator it is easy to verify that S
varies monotonically between 3 "^{1 / 2} = 0.57735 and 0.56606 as fw/r increases
from 0 to 1, provided β is always chosen such that it satisfies Eq. (5-6). This
very small variation of the scaling factor S is of no practical importance,

particularly as S is an easily measurable quantity. We are, of course, very glad that in return for being able to work with increased pulse widths we do not have to pay a high price in terms of an appreciably reduced scaling factor.

b. Flip Angle Errors Common to All Pulses; rf Inhomogeneity;

*Power Droop *

State-of-the-art multiple-pulse spectrometers usually have a means to
monitor the pulse power of the transmitter. By turning on the corresponding
knob one adjusts for the optimum flip angle ß*0**. Misadjustments lead to flip *
angle errors common to all pulses. Much more dangerous are, however, flip
angle errors that are not under control of the spectroscopist. (We contrast
the spectroscopist to the designer and builder of the spectrometer although
often they are the same person.) Such errors are caused by, e.g., all short- and
long-term variations and drifts of the transmitter power output. The notorious
power droop—the decrease of the transmitter power output during the course
of a long pulse train—is just a special case. One source of these troubles is
drifting and humming of dc power supplies. Another important source of
common flip angle errors is the unavoidable rf inhomogeneity, which makes
it impossible to adjust ß to its optimum value ß*0* for all volume elements of
the sample.

The discussion of common flip angle errors, rf inhomogeneity, and power droop is contained in a discussion of the dependence on ß of the NMR response of a solid sample to the particular multiple-pulse sequence at hand.

We shall study this dependence here for the WAHUHA sequence. Our main
concern is, of course, spectral resolution. There are two main reasons why a
deviation ε of β from its optimum value β*0* (by definition, ε = β — β*0**) affects *
the resolution attainable in multiple-pulse experiments:

(i) ε φ 0 leads to an incomplete suppression of dipolar interactions.

(ii) ε Φ 0 changes the scaling factor 5, and this leads via the rf inhomo- geneity to a deterioration of the spectral resolution.

There is also an indirect consequence of flip angle errors: The variation with
*β of the direction of B*eff in the toggling frame makes the resonance offset-
averaging mechanism dependent on β. We only mention this effect but do
not dwell upon it.

We are well prepared for our current subject: The dependence on ε of
ifD = <«^s£ular(0>av i^{s} contained in the coefficients of Table 5-2, and the
dependence on β of the scaling factor or, even more important, of

<7z(0>av ^ <^cs + ^)ff i^{s} contained in Eq. (5-7). For better perception of the
sensitivity to ε of J-fD, ^ cS + ^ff >^{ a n}d the scaling factor, we have compiled in
Table 5-3 the coefficients of the spin operator combinations occurring in JfD

and ^cs + ^>ff *^{n} terms of ε for ί*ψ**/τ -> 0. *

D. PULSE IMPERFECTIONS 111

TABLE 5-3

SENSITIVITY TO e OF THE COEFFICIENTS OF THE SPIN OPERATORS
OCCURRING IN <#D AND ^s+ « C f f^{a }

Operator

/,'/**

**w **

*Uh** ^{k }*
[/,'/.']+

[/*'//]+

[ W ] +

', 4
*h *

Coefficient in J^D or
sin^{2}e

sin^{4}e

— sin^{2}e(l + sin^{2}e)

— 4 cose sin^{2}e
2 cos^{2}e sine
2(1 + sin^{2}e) sine cose
cose

cos e(l —sine)
1 — sine + sin^{2}e

**JlQS ■+- ^****0****f f **

-> e^{2 }

**-► **

-> - e^{2 }
-> - 4 e^{2 }
- > 2 e

-►2e
-» 1 - e^{2}/2

for ίψ/τ -*· 0
+ 0(e^{4})

0(e^{4})
+ 0(e^{4})
+ 0 ( e^{4})
+ ö(e^{2})
+ 0(e^{3})
+ 0(e^{4})
-► l - ( e + i e^{2}) + 0(e^{3})
-> l - ( e - e^{2}) + 0(e^{3})

*a* [7p7g*]+ is an abbreviation for ± ( W + /e'/p*).

The coefficients of [/*'//] + , [/^Λ^ + , /y, and 7Z depend linearly on ε. It is via the corresponding terms in J^D and Jfcs + <#off that the inhomogeneity of the rf field, a power droop, and a misalignment of the transmitter power affect by far most strongly the spectral resolution in WAHUHA experiments.

Residual dipolar line broadening is expressed by the first two of these
terms, whereas the latter two lead—among other things—to line broadening
via the rf inhomogeneity. In the following subsection we shall see that by
combining variants of the WAHUHA sequence it is possible to eliminate all
coefficients of bilinear spin operators that are linear in I*y**. These are (what *
luck!) exactly those coefficients that are linear in ε and that are, as a result,
the most disturbing. Therefore, we shall not discuss residual dipolar line
broadening any further for the WAHUHA sequence.

On the other hand, it seems impossible to eliminate by this technique the coefficients linear in ε of linear spin operators, and thus the strongest direct line-broadening effect arising from rf inhomogeneity.

Garroway et al.* ^{90}* have shown, however, that by combining WAHUHA-
type cycles with cycles that contain 270° pulses in addition to 90° pulses it is
possible also to eliminate these terms. While the scheme has been proven to
work well for liquids it remains to be seen whether it is also useful for solids.

The direct rf inhomogeneity line-broadening mechanism in WAHUHA
experiments works as follows: In an rf coil there always exists a distribution
of the strength B*x* of the rf field, and consequently a distribution of ß over
the sample volume. Let us assume that this latter distribution is centered at
*ß**0**, that it is bell-shaped, and that its half-width at half-height is <ε*^{2}>^{1/2}. Con-
sider a narrow resonance line of a liquid sample, which in the ordinary NMR
spectrum is shifted off resonance by Δω. The center of the line appears in the

multiple-pulse spectrum at AcoS(ß*0**). Spins that are flipped by angles ß *
different from ß*0* "appear" in the multiple-pulse spectrum at AcoS(ß). S(ß) is
given by Eq. (5-8). By expressing S in terms of ε and assuming for simplicity
*ίψ/τ <ζ 1, which means β**0* « %π, we obtain

*AcoS(ß) -> AcoS(ß**0** + e) = Δω5(*π)(1+£β) = Aco(l/V3)(l.+ $e). (5-9) *
The distribution of the strength of the rf field is thus reflected directly in the
multiple-pulse lineshape of a liquid sample. The half-width at half-height of
the line is Aco(l/>/3)f <ε^{2}>^{1/2}. Note, in particular, that the width increases
linearly with the offset Δω. For WAHUHA experiments on liquids this is
typically the dominant line-broadening effect. For solids it is one line-
broadening mechanism among several others.

In order to get an idea of its practical importance let us choose <ε^{2}>^{1/2} =
0.052 ^= 3° and Δω = 2π x 2000 sec^{- 1}. These values lead to a line with a full
width at half-height of as much as 2π χ 80 Hz. Both input numbers of our
example are absolutely realistic: It requires special techniques and efforts to
wind small rf coils that produce rf fields substantially more homogeneous
over reasonably sized samples than specified by <ε^{2}>^{1/2} = 3°, although values
less than 1° have been reported.^{89} Δω = 2π χ 2000 Hz is a natural value to
choose for the center of proton spectra, which at ω0 = 2π χ 90 MHz have a
typical spread of 2πχ2250 Hz = 25 ppm. For^{ 1 9}F work even substantially
larger off-resonance shifts are often required.

c. Residual Dipolar Line Broadening; Compensation Schemes;

*MREV Eight-Pulse Sequence *

With regard to residual dipolar line broadening, we recognized those terms
in Tables 5-2 and 5-3 as the most dangerous ones that are linear in ε. As we
have shown in 1968^{51} it is possible to design compensation schemes that
eliminate these troubling terms. The most successful of them—up to this
date—seems to be the MREV eight-pulse sequence (see Fig. 5-1), which
consists of two subcycles: the first is a WAHUHA cycle; the second is again
a WAHUHA cycle, but the P*x* and P_*x* pulses are interchanged. The propa-
gation operator U*r{**(t) runs during the first MREV subcycle through exactly *
the same sequence of states as it does in a WAHUHA sequence. During the
second subcycle U*rf** (t) runs again through the same sequence of states, but *
the sign of I*x* is reversed everywhere.

Let us consider «?fD for the MREV sequence. The spin operator combinations involved are the same as for the WAHUHA sequence (see Table 5-2). The coefficients are one-half the sum of the respective coefficients from each sub- cycle. For the first one they are evidently identical with the corresponding coefficients of the WAHUHA cycle (see Table 5-2). We leave it as an easy exercise for the reader to show that the same coefficients are obtained again

**D. PULSE IMPERFECTIONS ** 113
for the second subcycle; however, the signs are reversed of all those coefficients
belonging to operators that contain I*y* linearly.

As a result, the MREV-coefficients of [I*x**%*^{k}*li + and U**y**%*^{k}*1 + vanish identi-*
cally, that is, regardless of the particular value of ß. A corresponding result is
obtained for J^*cs* and Jf*ofi*: The coefficient of I*y* vanishes identically.

These results have highly important consequences: All remaining co-
efficients in 3tf*O* now have two factors in common, namely,

**(** ^{i} **" ?)** ^{cosjS+} **? \** ^{sin} ** H** ^{ and cosj8} *** **

^{i}

^{cosjS+}

^{sin}

^{ and cosj8}

This means that there are now two possible choices of β for which ifD vanishes.

One is the same as for the WAHUHA cycle [see Eq. (5-6)] and the other is
*β = \π. Both choices coincide for ί**ψ**/τ = 0. *

Figure 5-7 shows how the surviving coefficients in «?fD vary with β for three choices of *W/T. The interesting point to note is that they all stay very small—

below 1%, say—for an appreciable range of β. This is true, in particular,
when ί*ψ**/τ is, on the one hand, nonzero so that the two zeros of JP**O* are well
separated, but when, on the other hand, it is small enough for the coefficients
to remain negligibly small between the zeros. Figure 5-7c shows that this is
definitely no longer the case for large pulse widths approaching the limiting
case fw -» τ. It is therefore not advisable to operate MREV sequences with
pulse widths approaching τ.

In summary, we may say that for small (but nonzero) pulse widths the sup- pression of dipolar spin-spin couplings by the MREV eight-pulse sequence is very insensitive to flip angle errors common to all pulses and therefore to misalignments, fluctuations, drifts, and a droop of the rf pulse power, and to the inhomogeneity of the rf field.

These properties of the MREV eight-pulse cycle—first predicted theoret-
ically by Mansfield^{83}—have been confirmed by specific experiments^{55} and
are confirmed by daily work in multiple-pulse laboratories—including ours—

all over the world.

What about the direct rf inhomogeneity line-broadening mechanism in
MREV experiments? For the WAHUHA sequence the coefficients in JP*CS *

and Jf*of{* of both I*y* and I*z* are linear in ε. Only the coefficient of I*y* is thrown out
with the MREV sequence, so that we are left with one coefficient linear in ε.

The counterpart of Eq. (5-9) is for the MREV sequence

**fori****w****/T->0 **

*AcoS(ß) - AcoS(ß**0** + e) = ΔωΑ(*π)(1 +$e) = Δω^V2(l +$e). *

(5-10) By comparing Eqs. (5-9) and (5-10) we see that the direct rf inhomogeneity line-broadening mechanism is almost as effective for the MREV as for the

FIG. 5-7. Average dipolar Hamiltonian *^D for MREV sequence versus flip angle ß. The
curves show the coefficients of iUx%^{k}* + h%*^{K}*), ; /*'/**, ; and 7*z72k, - - ; for
three different choices of /_{W}/T. The coefficient of /,'//, though nonzero for βφ\π and /?_{0},
is always negligibly small. Note that the horizontal and vertical scales, respectively, are
equal for /W/T = 0 and 0.2, but much larger for ίγ,/τ = 1. Note in particular the range of β
for which all coefficients stay smaller than, e.g., 10"^{ 2}. For /W/T = 0.2 this range exceeds 9°.

WAHUHA sequence. Carefully optimizing the homogeneity of the rf field is therefore mandatory if one wishes to exploit fully the capability of suppressing dipolar spin-spin interactions in solids with any of these multiple-pulse sequences.

2. FLIP ANGLE AND PHASE ERRORS OF INDIVIDUAL PULSES;

PHASE TRANSIENTS

We shall discuss this class of pulse imperfections together. Their common
characteristic feature is that they destroy the cyclic property of multiple-pulse
sequences. This means the propagation operator U*rf**(t) does not return to *
unity after a full cycle in the presence of these imperfections. While this seems